\newproblem{lay:5_5_1}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 5.5.1}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let the matrix $A=\begin{pmatrix}1 & -2 \\ 1 & 3\end{pmatrix}$ act on $\mathbb{C}^2$. Find the eigenvalues and a basis for each of the eigenspace in $\mathbb{C}^2$.
}{
  % Solution
	The eigenvalues are the solutions of the characteristic equation
	\begin{center}
		$|A-\lambda I|=0$ \\
		$\left|\begin{array}{cc}1-\lambda & -2 \\ 1 & 3-\lambda\end{array}\right|=(1-\lambda)(3-\lambda)+2=(\lambda-(2+i))(\lambda-(2-i))=0$
	\end{center}
	The two eigenvalues are complex. Let's find now a basis for each one of the eigenspaces.\\

	\underline{$\lambda=2+i$}\\
	We need to solve the homogeneous equation system $(A-\lambda I)\mathbf{v}=\mathbf{0}$
	\begin{center}
		$\begin{pmatrix}1-(2+i) & -2 \\ 1 & 3-(2+i)\end{pmatrix}\mathbf{v}=\mathbf{0}$
	\end{center}
	We use the augmented matrix below
	\begin{center}
		$\left(\begin{array}{cc|c}-1-i & -2 & 0 \\ 1 & 1-i & 0\end{array}\right) \sim \left(\begin{array}{cc|c}1 & 1-i & 0 \\ 0 & 0 & 0\end{array}\right)$
	\end{center}
	All vectors in this eigenspace are of the form $\mathbf{v}=((-1+i)x_2,x_2)\quad x_2\in\mathbb{R}$. One of its bases is $\{(-1+i,1)\}$\\

	\underline{$\lambda=2-i$}\\
	We need to solve the homogeneous equation system $(A-\lambda I)\mathbf{v}=\mathbf{0}$
	\begin{center}
		$\begin{pmatrix}1-(2-i) & -2 \\ 1 & 3-(2-i)\end{pmatrix}\mathbf{v}=\mathbf{0}$
	\end{center}
	We use the augmented matrix below
	\begin{center}
		$\left(\begin{array}{cc|c}-1+i & -2 & 0 \\ 1 & 1+i & 0\end{array}\right) \sim \left(\begin{array}{cc|c}1 & 1+i & 0 \\ 0 & 0 & 0\end{array}\right)$
	\end{center}
	All vectors in this eigenspace are of the form $\mathbf{v}=((-1-i)x_2,x_2)\quad x_2\in\mathbb{R}$. One of its bases is $\{(-1-i,1)\}$
	
	In fact this is a general result, if $\lambda$ and $\lambda^*$ are two complex conjugate eigenvalues, then their corresponding bases are also related by 
	a complex conjugate operation.
}
\useproblem{lay:5_5_1}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
